Optimal. Leaf size=148 \[ \frac {\sqrt {\frac {6-\left (7-\sqrt {73}\right ) x^2}{6-\left (7+\sqrt {73}\right ) x^2}} \sqrt {\left (7+\sqrt {73}\right ) x^2-6} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{73} x}{\sqrt {\left (7+\sqrt {73}\right ) x^2-6}}\right )|\frac {1}{146} \left (73+7 \sqrt {73}\right )\right )}{2 \sqrt {3} \sqrt [4]{73} \sqrt {\frac {1}{6-\left (7+\sqrt {73}\right ) x^2}} \sqrt {2 x^4+7 x^2-3}} \]
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Rubi [A] time = 0.03, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1098} \[ \frac {\sqrt {\frac {6-\left (7-\sqrt {73}\right ) x^2}{6-\left (7+\sqrt {73}\right ) x^2}} \sqrt {\left (7+\sqrt {73}\right ) x^2-6} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{73} x}{\sqrt {\left (7+\sqrt {73}\right ) x^2-6}}\right )|\frac {1}{146} \left (73+7 \sqrt {73}\right )\right )}{2 \sqrt {3} \sqrt [4]{73} \sqrt {\frac {1}{6-\left (7+\sqrt {73}\right ) x^2}} \sqrt {2 x^4+7 x^2-3}} \]
Antiderivative was successfully verified.
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Rule 1098
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3+7 x^2+2 x^4}} \, dx &=\frac {\sqrt {\frac {6-\left (7-\sqrt {73}\right ) x^2}{6-\left (7+\sqrt {73}\right ) x^2}} \sqrt {-6+\left (7+\sqrt {73}\right ) x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{73} x}{\sqrt {-6+\left (7+\sqrt {73}\right ) x^2}}\right )|\frac {1}{146} \left (73+7 \sqrt {73}\right )\right )}{2 \sqrt {3} \sqrt [4]{73} \sqrt {\frac {1}{6-\left (7+\sqrt {73}\right ) x^2}} \sqrt {-3+7 x^2+2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 80, normalized size = 0.54 \[ -\frac {i \sqrt {-4 x^4-14 x^2+6} F\left (i \sinh ^{-1}\left (\frac {2 x}{\sqrt {7+\sqrt {73}}}\right )|\frac {1}{12} \left (-61-7 \sqrt {73}\right )\right )}{\sqrt {\sqrt {73}-7} \sqrt {2 x^4+7 x^2-3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {2 \, x^{4} + 7 \, x^{2} - 3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, x^{4} + 7 \, x^{2} - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 84, normalized size = 0.57 \[ \frac {6 \sqrt {-\left (\frac {7}{6}-\frac {\sqrt {73}}{6}\right ) x^{2}+1}\, \sqrt {-\left (\frac {\sqrt {73}}{6}+\frac {7}{6}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {42-6 \sqrt {73}}\, x}{6}, \frac {7 i \sqrt {6}}{12}+\frac {i \sqrt {438}}{12}\right )}{\sqrt {42-6 \sqrt {73}}\, \sqrt {2 x^{4}+7 x^{2}-3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, x^{4} + 7 \, x^{2} - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {2\,x^4+7\,x^2-3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 x^{4} + 7 x^{2} - 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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